derivative of 
sin
⁡
(
𝑥
3
)
sin(x 
3
 )

To find the derivative of \( \sin(x^3) \), we will use the chain rule. 

### Step-by-Step Solution:

1. **Identify the outer function**: The outer function is \( \sin(u) \), where \( u = x^3 \).
2. **Differentiate the outer function**: The derivative of \( \sin(u) \) with respect to \( u \) is \( \cos(u) \).
3. **Identify the inner function**: The inner function is \( u = x^3 \).
4. **Differentiate the inner function**: The derivative of \( x^3 \) with respect to \( x \) is \( 3x^2 \).
5. **Apply the chain rule**: Multiply the derivative of the outer function by the derivative of the inner function.

The derivative of \( \sin(x^3) \) with respect to \( x \) is:

\[
\frac{d}{dx} \left( \sin(x^3) \right) = \cos(x^3) \cdot 3x^2
\]

Thus, the derivative is:

\[
3x^2 \cdot \cos(x^3)
\]

Would you like more details or have any questions?

Here are some related questions:
1. What is the derivative of \( \cos(x^3) \)?
2. How would you find the derivative of \( \sin(x^2 + 1) \)?
3. What is the second derivative of \( \sin(x^3) \)?
4. How do you apply the chain rule for composite functions?
5. How do you find the derivative of \( \sin(kx) \), where \( k \) is a constant?

**Tip**: Always identify the outer and inner functions when using the chain rule for differentiation.

Learn how to find the derivative of sin(x^3) using the chain rule. This calculus problem involves identifying the outer and inner functions and applying the chain rule for differentiation. Suitable for students in Grades 11-12.

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